Euler's formula
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Euler's formula, named after
Leonhard Euler Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries in ma ...
, is a
mathematical Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
formula In science, a formula is a concise way of expressing information symbolically, as in a mathematical formula or a ''chemical formula''. The informal use of the term ''formula'' in science refers to the general construct of a relationship betwee ...
in
complex analysis Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates Function (mathematics), functions of complex numbers. It is helpful in many branches of mathemati ...
that establishes the fundamental relationship between the
trigonometric functions In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in all ...
and the
complex Complex commonly refers to: * Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe ** Complex system, a system composed of many components which may interact with each ...
exponential function The exponential function is a mathematical function denoted by f(x)=\exp(x) or e^x (where the argument is written as an exponent). Unless otherwise specified, the term generally refers to the positive-valued function of a real variable, a ...
. Euler's formula states that for any
real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...
 : e^ = \cos x + i\sin x, where is the base of the natural logarithm, is the
imaginary unit The imaginary unit or unit imaginary number () is a solution to the quadratic equation x^2+1=0. Although there is no real number with this property, can be used to extend the real numbers to what are called complex numbers, using addition an ...
, and and are the
trigonometric functions In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in all ...
cosine and
sine In mathematics, sine and cosine are trigonometric functions of an angle. The sine and cosine of an acute angle are defined in the context of a right triangle: for the specified angle, its sine is the ratio of the length of the side that is oppo ...
respectively. This complex exponential function is sometimes denoted ("cosine plus i sine"). The formula is still valid if is a
complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...
, and so some authors refer to the more general complex version as Euler's formula. Euler's formula is ubiquitous in mathematics, physics, and engineering. The physicist
Richard Feynman Richard Phillips Feynman (; May 11, 1918 – February 15, 1988) was an American theoretical physicist, known for his work in the path integral formulation of quantum mechanics, the theory of quantum electrodynamics, the physics of the superflu ...
called the equation "our jewel" and "the most remarkable formula in mathematics". When , Euler's formula may be rewritten as , which is known as
Euler's identity In mathematics, Euler's identity (also known as Euler's equation) is the equality e^ + 1 = 0 where : is Euler's number, the base of natural logarithms, : is the imaginary unit, which by definition satisfies , and : is pi, the ratio of the circ ...
.


History

In 1714, the English mathematician
Roger Cotes Roger Cotes (10 July 1682 – 5 June 1716) was an English mathematician, known for working closely with Isaac Newton by proofreading the second edition of his famous book, the '' Principia'', before publication. He also invented the quadratur ...
presented a geometrical argument that can be interpreted (after correcting a misplaced factor of \sqrt) as: ix = \ln(\cos x + i\sin x). Exponentiating this equation yields Euler's formula. Note that the logarithmic statement is not universally correct for complex numbers, since a complex logarithm can have infinitely many values, differing by multiples of . Around 1740
Leonhard Euler Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries in ma ...
turned his attention to the exponential function and derived the equation named after him by comparing the series expansions of the exponential and trigonometric expressions. The formula was first published in 1748 in his foundational work ''
Introductio in analysin infinitorum ''Introductio in analysin infinitorum'' (Latin: ''Introduction to the Analysis of the Infinite'') is a two-volume work by Leonhard Euler which lays the foundations of mathematical analysis. Written in Latin and published in 1748, the ''Introducti ...
''.
Johann Bernoulli Johann Bernoulli (also known as Jean or John; – 1 January 1748) was a Swiss mathematician and was one of the many prominent mathematicians in the Bernoulli family. He is known for his contributions to infinitesimal calculus and educating L ...
had found that \frac = \frac 1 2 \left( \frac + \frac\right). And since \int \frac = \frac \ln(1 + ax) + C, the above equation tells us something about complex logarithms by relating natural logarithms to imaginary (complex) numbers. Bernoulli, however, did not evaluate the integral. Bernoulli's correspondence with Euler (who also knew the above equation) shows that Bernoulli did not fully understand complex logarithms. Euler also suggested that complex logarithms can have infinitely many values. The view of complex numbers as points in the
complex plane In mathematics, the complex plane is the plane formed by the complex numbers, with a Cartesian coordinate system such that the -axis, called the real axis, is formed by the real numbers, and the -axis, called the imaginary axis, is formed by the ...
was described about 50 years later by
Caspar Wessel Caspar Wessel (8 June 1745, Vestby – 25 March 1818, Copenhagen) was a Danish– Norwegian mathematician and cartographer. In 1799, Wessel was the first person to describe the geometrical interpretation of complex numbers as points in the comp ...
.


Definitions of complex exponentiation

The exponential function for real values of may be defined in a few different equivalent ways (see
Characterizations of the exponential function In mathematics, the exponential function can be characterized in many ways. The following characterizations (definitions) are most common. This article discusses why each characterization makes sense, and why the characterizations are independent o ...
). Several of these methods may be directly extended to give definitions of for complex values of simply by substituting in place of and using the complex algebraic operations. In particular we may use any of the three following definitions, which are equivalent. From a more advanced perspective, each of these definitions may be interpreted as giving the unique
analytic continuation In complex analysis, a branch of mathematics, analytic continuation is a technique to extend the domain of definition of a given analytic function. Analytic continuation often succeeds in defining further values of a function, for example in a new ...
of to the complex plane.


Differential equation definition

The exponential function z \mapsto e^z is the unique
differentiable function In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non-vertical tangent line at each interior point in its ...
of a
complex variable Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is helpful in many branches of mathematics, including algebraic ...
for which the derivative equals the function \frac = e^z and e^0 = 1.


Power series definition

For complex e^z = 1 + \frac + \frac + \frac + \cdots = \sum_^ \frac. Using the
ratio test In mathematics, the ratio test is a test (or "criterion") for the convergence of a series :\sum_^\infty a_n, where each term is a real or complex number and is nonzero when is large. The test was first published by Jean le Rond d'Alembert a ...
, it is possible to show that this
power series In mathematics, a power series (in one variable) is an infinite series of the form \sum_^\infty a_n \left(x - c\right)^n = a_0 + a_1 (x - c) + a_2 (x - c)^2 + \dots where ''an'' represents the coefficient of the ''n''th term and ''c'' is a const ...
has an infinite
radius of convergence In mathematics, the radius of convergence of a power series is the radius of the largest disk at the center of the series in which the series converges. It is either a non-negative real number or \infty. When it is positive, the power series co ...
and so defines for all complex .


Limit definition

For complex e^z = \lim_ \left(1+\frac\right)^n. Here, is restricted to positive integers, so there is no question about what the power with exponent means.


Proofs

Various proofs of the formula are possible.


Using differentiation

This proof shows that the quotient of the trigonometric and exponential expressions is the constant function one, so they must be equal (the exponential function is never zero, Theorem 1.42 so this is permitted). Consider the function f(\theta) = \frac = e^ \left(\cos\theta + i \sin\theta\right) for real . Differentiating gives by the
product rule In calculus, the product rule (or Leibniz rule or Leibniz product rule) is a formula used to find the derivatives of products of two or more functions. For two functions, it may be stated in Lagrange's notation as (u \cdot v)' = u ' \cdot v + ...
f'(\theta) = e^ \left(i\cos\theta - \sin\theta\right) - ie^ \left(\cos\theta + i\sin\theta\right) = 0 Thus, is a constant. Since , then for all real , and thus e^ = \cos\theta + i\sin\theta.


Using power series

Here is a proof of Euler's formula using power-series expansions, as well as basic facts about the powers of : \begin i^0 &= 1, & i^1 &= i, & i^2 &= -1, & i^3 &= -i, \\ i^4 &= 1, & i^5 &= i, & i^6 &= -1, & i^7 &= -i \\ &\vdots & &\vdots & &\vdots & &\vdots \end Using now the power-series definition from above, we see that for real values of \begin e^ &= 1 + ix + \frac + \frac + \frac + \frac + \frac + \frac + \frac + \cdots \\ pt &= 1 + ix - \frac - \frac + \frac + \frac - \frac - \frac + \frac + \cdots \\ pt &= \left( 1 - \frac + \frac - \frac + \frac - \cdots \right) + i\left( x - \frac + \frac - \frac + \cdots \right) \\ pt &= \cos x + i\sin x , \end where in the last step we recognize the two terms are the
Maclaurin series Maclaurin or MacLaurin is a surname. Notable people with the surname include: * Colin Maclaurin (1698–1746), Scottish mathematician * Normand MacLaurin (1835–1914), Australian politician and university administrator * Henry Normand MacLaurin ( ...
for and . The rearrangement of terms is justified because each series is
absolutely convergent In mathematics, an infinite series of numbers is said to converge absolutely (or to be absolutely convergent) if the sum of the absolute values of the summands is finite. More precisely, a real or complex series \textstyle\sum_^\infty a_n is s ...
.


Using polar coordinates

Another proof Second proof on page. is based on the fact that all complex numbers can be expressed in
polar coordinates In mathematics, the polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by a distance from a reference point and an angle from a reference direction. The reference point (analogous to the or ...
. Therefore, for some and depending on , e^ = r \left(\cos \theta + i \sin \theta\right). No assumptions are being made about and ; they will be determined in the course of the proof. From any of the definitions of the exponential function it can be shown that the derivative of is . Therefore, differentiating both sides gives i e ^ = \left(\cos \theta + i \sin \theta\right) \frac + r \left(-\sin \theta + i \cos \theta\right) \frac. Substituting for and equating real and imaginary parts in this formula gives and . Thus, is a constant, and is for some constant . The initial values and come from , giving and . This proves the formula e^ = 1(\cos x +i \sin x) = \cos x + i \sin x.


Applications


Applications in complex number theory


Interpretation of the formula

This formula can be interpreted as saying that the function is a
unit complex number In mathematics, the circle group, denoted by \mathbb T or \mathbb S^1, is the multiplicative group of all complex numbers with absolute value 1, that is, the unit circle in the complex plane or simply the unit complex numbers. \mathbb T = \. ...
, i.e., it traces out the
unit circle In mathematics, a unit circle is a circle of unit radius—that is, a radius of 1. Frequently, especially in trigonometry, the unit circle is the circle of radius 1 centered at the origin (0, 0) in the Cartesian coordinate system in the Eucl ...
in the
complex plane In mathematics, the complex plane is the plane formed by the complex numbers, with a Cartesian coordinate system such that the -axis, called the real axis, is formed by the real numbers, and the -axis, called the imaginary axis, is formed by the ...
as ranges through the real numbers. Here is the
angle In Euclidean geometry, an angle is the figure formed by two Ray (geometry), rays, called the ''Side (plane geometry), sides'' of the angle, sharing a common endpoint, called the ''vertex (geometry), vertex'' of the angle. Angles formed by two ...
that a line connecting the origin with a point on the unit circle makes with the
positive real axis In mathematics, the set of positive real numbers, \R_ = \left\, is the subset of those real numbers that are greater than zero. The non-negative real numbers, \R_ = \left\, also include zero. Although the symbols \R_ and \R^ are ambiguously used fo ...
, measured counterclockwise and in
radian The radian, denoted by the symbol rad, is the unit of angle in the International System of Units (SI) and is the standard unit of angular measure used in many areas of mathematics. The unit was formerly an SI supplementary unit (before that c ...
s. The original proof is based on the
Taylor series In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor serie ...
expansions of the
exponential function The exponential function is a mathematical function denoted by f(x)=\exp(x) or e^x (where the argument is written as an exponent). Unless otherwise specified, the term generally refers to the positive-valued function of a real variable, a ...
(where is a complex number) and of and for real numbers (see below). In fact, the same proof shows that Euler's formula is even valid for all ''complex'' numbers . A point in the
complex plane In mathematics, the complex plane is the plane formed by the complex numbers, with a Cartesian coordinate system such that the -axis, called the real axis, is formed by the real numbers, and the -axis, called the imaginary axis, is formed by the ...
can be represented by a complex number written in cartesian coordinates. Euler's formula provides a means of conversion between cartesian coordinates and
polar coordinates In mathematics, the polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by a distance from a reference point and an angle from a reference direction. The reference point (analogous to the or ...
. The polar form simplifies the mathematics when used in multiplication or powers of complex numbers. Any complex number , and its complex conjugate, , can be written as \begin z &= x + iy = , z, (\cos \varphi + i\sin \varphi) = r e^, \\ \bar &= x - iy = , z, (\cos \varphi - i\sin \varphi) = r e^, \end where * is the real part, * is the imaginary part, * is the
magnitude Magnitude may refer to: Mathematics *Euclidean vector, a quantity defined by both its magnitude and its direction *Magnitude (mathematics), the relative size of an object *Norm (mathematics), a term for the size or length of a vector *Order of ...
of and *. is the
argument An argument is a statement or group of statements called premises intended to determine the degree of truth or acceptability of another statement called conclusion. Arguments can be studied from three main perspectives: the logical, the dialectic ...
of , i.e., the angle between the ''x'' axis and the vector ''z'' measured counterclockwise in
radian The radian, denoted by the symbol rad, is the unit of angle in the International System of Units (SI) and is the standard unit of angular measure used in many areas of mathematics. The unit was formerly an SI supplementary unit (before that c ...
s, which is defined
up to Two Mathematical object, mathematical objects ''a'' and ''b'' are called equal up to an equivalence relation ''R'' * if ''a'' and ''b'' are related by ''R'', that is, * if ''aRb'' holds, that is, * if the equivalence classes of ''a'' and ''b'' wi ...
addition of . Many texts write instead of , but the first equation needs adjustment when . This is because for any real and , not both zero, the angles of the vectors and differ by radians, but have the identical value of .


Use of the formula to define the logarithm of complex numbers

Now, taking this derived formula, we can use Euler's formula to define the
logarithm In mathematics, the logarithm is the inverse function to exponentiation. That means the logarithm of a number  to the base  is the exponent to which must be raised, to produce . For example, since , the ''logarithm base'' 10 o ...
of a complex number. To do this, we also use the definition of the logarithm (as the inverse operator of exponentiation): a = e^, and that e^a e^b = e^, both valid for any complex numbers and . Therefore, one can write: z = \left, z\ e^ = e^ e^ = e^ for any . Taking the logarithm of both sides shows that \ln z = \ln \left, z\ + i \varphi, and in fact, this can be used as the definition for the complex logarithm. The logarithm of a complex number is thus a
multi-valued function In mathematics, a multivalued function, also called multifunction, many-valued function, set-valued function, is similar to a function, but may associate several values to each input. More precisely, a multivalued function from a domain to ...
, because is multi-valued. Finally, the other exponential law \left(e^a\right)^k = e^, which can be seen to hold for all integers , together with Euler's formula, implies several
trigonometric identities In trigonometry, trigonometric identities are equalities that involve trigonometric functions and are true for every value of the occurring variables for which both sides of the equality are defined. Geometrically, these are identities involvin ...
, as well as
de Moivre's formula In mathematics, de Moivre's formula (also known as de Moivre's theorem and de Moivre's identity) states that for any real number and integer it holds that :\big(\cos x + i \sin x\big)^n = \cos nx + i \sin nx, where is the imaginary unit (). ...
.


Relationship to trigonometry

Euler's formula, the definitions of the trigonometric functions and the standard identities for exponentials are sufficient to easily derive most trigonometric identities. It provides a powerful connection between
analysis Analysis ( : analyses) is the process of breaking a complex topic or substance into smaller parts in order to gain a better understanding of it. The technique has been applied in the study of mathematics and logic since before Aristotle (38 ...
and
trigonometry Trigonometry () is a branch of mathematics that studies relationships between side lengths and angles of triangles. The field emerged in the Hellenistic world during the 3rd century BC from applications of geometry to astronomical studies. T ...
, and provides an interpretation of the sine and cosine functions as
weighted sum A weight function is a mathematical device used when performing a sum, integral, or average to give some elements more "weight" or influence on the result than other elements in the same set. The result of this application of a weight function is ...
s of the exponential function: \begin \cos x &= \operatorname \left(e^\right) =\frac, \\ \sin x &= \operatorname \left(e^\right) =\frac. \end The two equations above can be derived by adding or subtracting Euler's formulas: \begin e^ &= \cos x + i \sin x, \\ e^ &= \cos(- x) + i \sin(- x) = \cos x - i \sin x \end and solving for either cosine or sine. These formulas can even serve as the definition of the trigonometric functions for complex arguments . For example, letting , we have: \begin \cos iy &= \frac = \cosh y, \\ \sin iy &= \frac = \fraci = i\sinh y. \end Complex exponentials can simplify trigonometry, because they are easier to manipulate than their sinusoidal components. One technique is simply to convert sinusoids into equivalent expressions in terms of exponentials. After the manipulations, the simplified result is still real-valued. For example: \begin \cos x \cos y &= \frac \cdot \frac \\ &= \frac 1 2 \cdot \frac \\ &= \frac 1 2 \bigg( \underbrace_ + \underbrace_ \bigg). \end Another technique is to represent the sinusoids in terms of the
real part In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...
of a complex expression and perform the manipulations on the complex expression. For example: \begin \cos nx &= \operatorname \left(e^\right) \\ &= \operatorname \left( e^\cdot e^ \right) \\ &= \operatorname \Big( e^\cdot \big(\underbrace_ - e^\big) \Big) \\ &= \operatorname \left( e^\cdot 2\cos x - e^ \right) \\ &= \cos n-1)x\cdot \cos x- \cos n-2)x \end This formula is used for recursive generation of for integer values of and arbitrary (in radians).


Topological interpretation

In the language of
topology In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformations, such ...
, Euler's formula states that the imaginary exponential function t \mapsto e^ is a (
surjective In mathematics, a surjective function (also known as surjection, or onto function) is a function that every element can be mapped from element so that . In other words, every element of the function's codomain is the image of one element of i ...
)
morphism In mathematics, particularly in category theory, a morphism is a structure-preserving map from one mathematical structure to another one of the same type. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphisms a ...
of
topological group In mathematics, topological groups are logically the combination of groups and topological spaces, i.e. they are groups and topological spaces at the same time, such that the continuity condition for the group operations connects these two str ...
s from the real line \mathbb R to the unit circle \mathbb S^1. In fact, this exhibits \mathbb R as a
covering space A covering of a topological space X is a continuous map \pi : E \rightarrow X with special properties. Definition Let X be a topological space. A covering of X is a continuous map : \pi : E \rightarrow X such that there exists a discrete spa ...
of \mathbb S^1. Similarly,
Euler's identity In mathematics, Euler's identity (also known as Euler's equation) is the equality e^ + 1 = 0 where : is Euler's number, the base of natural logarithms, : is the imaginary unit, which by definition satisfies , and : is pi, the ratio of the circ ...
says that the
kernel Kernel may refer to: Computing * Kernel (operating system), the central component of most operating systems * Kernel (image processing), a matrix used for image convolution * Compute kernel, in GPGPU programming * Kernel method, in machine learnin ...
of this map is \tau \mathbb Z, where \tau = 2\pi. These observations may be combined and summarized in the commutative diagram below:


Other applications

In
differential equation In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, an ...
s, the function is often used to simplify solutions, even if the final answer is a real function involving sine and cosine. The reason for this is that the exponential function is the
eigenfunction In mathematics, an eigenfunction of a linear operator ''D'' defined on some function space is any non-zero function f in that space that, when acted upon by ''D'', is only multiplied by some scaling factor called an eigenvalue. As an equation, th ...
of the operation of differentiation. In
electrical engineering Electrical engineering is an engineering discipline concerned with the study, design, and application of equipment, devices, and systems which use electricity, electronics, and electromagnetism. It emerged as an identifiable occupation in the l ...
,
signal processing Signal processing is an electrical engineering subfield that focuses on analyzing, modifying and synthesizing ''signals'', such as audio signal processing, sound, image processing, images, and scientific measurements. Signal processing techniq ...
, and similar fields, signals that vary periodically over time are often described as a combination of sinusoidal functions (see
Fourier analysis In mathematics, Fourier analysis () is the study of the way general functions may be represented or approximated by sums of simpler trigonometric functions. Fourier analysis grew from the study of Fourier series, and is named after Josep ...
), and these are more conveniently expressed as the sum of exponential functions with imaginary exponents, using Euler's formula. Also,
phasor analysis In physics and engineering, a phasor (a portmanteau of phase vector) is a complex number representing a sinusoidal function whose amplitude (''A''), angular frequency (''ω''), and initial phase (''θ'') are time-invariant. It is related to a ...
of circuits can include Euler's formula to represent the impedance of a capacitor or an inductor. In the
four-dimensional space A four-dimensional space (4D) is a mathematical extension of the concept of three-dimensional or 3D space. Three-dimensional space is the simplest possible abstraction of the observation that one only needs three numbers, called ''dimensions'', ...
of
quaternion In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. Hamilton defined a quatern ...
s, there is a
sphere A sphere () is a Geometry, geometrical object that is a solid geometry, three-dimensional analogue to a two-dimensional circle. A sphere is the Locus (mathematics), set of points that are all at the same distance from a given point in three ...
of
imaginary unit The imaginary unit or unit imaginary number () is a solution to the quadratic equation x^2+1=0. Although there is no real number with this property, can be used to extend the real numbers to what are called complex numbers, using addition an ...
s. For any point on this sphere, and a real number, Euler's formula applies: \exp xr = \cos x + r \sin x, and the element is called a
versor In mathematics, a versor is a quaternion of norm one (a ''unit quaternion''). The word is derived from Latin ''versare'' = "to turn" with the suffix ''-or'' forming a noun from the verb (i.e. ''versor'' = "the turner"). It was introduced by Will ...
in quaternions. The set of all versors forms a 3-sphere in the 4-space.


See also

*
Complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...
*
Euler's identity In mathematics, Euler's identity (also known as Euler's equation) is the equality e^ + 1 = 0 where : is Euler's number, the base of natural logarithms, : is the imaginary unit, which by definition satisfies , and : is pi, the ratio of the circ ...
*
Integration using Euler's formula In integral calculus, Euler's formula for complex numbers may be used to evaluate integrals involving trigonometric functions. Using Euler's formula, any trigonometric function may be written in terms of complex exponential functions, namely e^ a ...
* *
List of things named after Leonhard Euler 200px, Leonhard Euler (1707–1783) In mathematics and physics, many topics are named in honor of Swiss mathematician Leonhard Euler (1707–1783), who made many important discoveries and innovations. Many of these items named after Euler inclu ...


References


Further reading

* *


External links


Elements of Algebra
{{Leonhard Euler Theorems in complex analysis Articles containing proofs Analysis E (mathematical constant) Trigonometry Leonhard Euler